An extended BFM model for simulation of copolymers at an interface

COMPUTATIONAL MATERIALS SCIENCE ELSEVIER

Computational Materials Science 10 (1998) 175-179

An extended BFM model for simulation of copolymers at an interface E. James *, C.C. Matthai Department of Physics and Astronomy, Universify of Wales, Cardifi Card@CF2 SYB, UK

Abstract We have extended the original bond fluctuation model (BFM) of Carmesin and Kremer [Macromolecules 12 (1988) 28191 so that it allows for a threefold an increase in the possible bonding configurations. This appears to be computationally more efficient in determining equilibrium configurations. We have used this modified BFM to simulate random copolymers at the interface of two good solvents. Copyright 0 1998 Elsevier Science B.V.

1. Introduction

In recent years there has been an increase in computer simulation studies of polymer systems. Because of the different time scales in the motions of polymers, it is virtually impossible to simulate realistic long chain polymers even with simple interaction potentials. Much work has therefore concentrated on different aspects of the motion and coarse-grained models, where the details of the chemical structure on the monomer length scale are omitted, have formed the basis of these simulations. In these approaches, a monomer unit can be comprised of many molecules and bonds. Lattice models, in which the monomer units are associated with lattice sites, are computationally more efficient than continuous space models and even though they are slightly less realistic, the results obtained using these models are of interest. Also, the Monte Carlo method is well suited to performing simulations on lattice models. The bond fluctuating model (BFM) advanced by Carmesin and

* Corresponding author. E-mail: [email protected].

Kremer [l] and Deutsch and Binder [2] has all the advantages of simple lattice models but in allowing for a greater choice of bond vectors is able to give a better description of polymer dynamics enabling it to be used in the study of a wide variety of polymer systems. The BFM has the advantage of using the efficiency of lattice models and adding the flexibility of the continuum models. In this paper, we report on how the BFM may be extended to give even more flexibility without losing any efficiency. The extension is described in the next section which is followed by the application of the model to study copolymers at an interface.

2. The extended BFlU model In the original BFM, polymers of length n are represented by beads (monomers) connected by (n - 1) bonds. The relationship between the model and a real polymer is that the beads and bonds represent Kuhn segments which are a sequence of a few tens of monomers of the real chain. These segments should be

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176

E. James, C.C. Matrhai/Computatiorlul

Materials

Science

10 (1998)

17S-179

allowed to have a range of lengths to allow for kinking

where P(x. I’, z) stands for the set of all permuta-

and stretching of the real chain, and there should be

tions and sign combinations

very little restriction

results in 318 possible bonds, which is almost three

two successive

on the possible

angle between

segments. The polymer is mapped on

to a simple cubic lattice and each monomer occupies a (2 x 2 x 2) cube containing sites. The cubes of the neighbouring not allowed

or bead

eight lattice monomers

are

to overlap which in effect models ex-

cluded volume constraints.

Within this framework,

times that in the original

of (kx, fy,

&z). This

model. With this modifi-

cation and leaving all the rules of moves the same, we tested

the model

fore performing

for computer

the simulations

efficiency

of polymers

beat an

interface.

it

is possible to have many more bond lengths and bond angles than is possible in a simple single site lattice model.

The initial

polymer

configuration

3. Single chain test

is set up

by restricting the bonds to an allowed set of vectors which are obtained by requiring that no two bonds to cross each other and that all bonds are reachable from each other by successive bead jumps. The Monte Carlo method is used to perform the simulations. Moves are made by selecting a monomer and moving

As a test of the model for computer efficiency, we considered

a single free polymer

chain of 64 units

and performed the MC simulations to equilibrate the chain using both the original and extended BFM. The chain was allowed to evolve for 200000

moves and

it one lattice spacing in the direction of a lattice basis

the bond angles between successive segments and the bond lengths were recorded every 10 moves. The re-

vector. A move is accepted if it does not touch or intersect another monomer and if any bonds associated

respectively.

with the monomer remain in the allowed bond set.

there are many more structural possibilities.

sulting distributions

are displayed

in Figs. I and 2,

It can be seen that with the new model, The num-

Although this model has proved to be very success-

ber of possible bond lengths increases from 5 to 13 and

ful in simulating many polymer systems, it is of interest to see if this flexibility in bonding configurations

the angles from 100 to 600. This is only to be expected in view of the finer lattice mesh relative to the poly-

and dynamics can be further increased without affecting the computational efficiency. The obvious way to

mer chain. What was interesting, however, was that the acceptance ratio for moves in the old model was only 0.27 as compared to 0.37 with the modification.

do this is to increase the size of the unit cell occupied by the monomer unit. Of course, in the limiting case of the lattice spacing becoming infinitely small,

The consequence

we recover the continuum limit but there will be much reduced efficiency. We therefore propose an extension to the BFM in which the centre of each monomer is

bond vectors for each move, because the acceptance librium configuration

situated on a lattice site (unlike in the original model)

this advantage to become more pronounced

and occupies a (3 x 3 x 3) cube containing 27 lattice sites. Then, by following the rules of Deutsch and

systems.

Binder [2] for determining the allowed set of vectors for the bonds and using their notation, we find that these are given by

B = P(3,0,0) u P(3,2,

U P(3, 1,0) u P(3, 1, 1) u P(3,2,0) 1) u P(3,2,2)

u P(3,3,0)

u P(3.3.2)

u P(4,0,0)

u P(4. 1,O) u P(4, 1, 1) u P(4,2,0)

u P(4,2,2)

u P(5,0,0)

u P(5, 1,0)

of this was that although marginally

more time is spent searching through all the allowed ratio is greater, the overall time taken to reach an equi-

4. Simulation interface

is actually reduced. We expect

of a random copolymer

for denser

at an

Having demonstrated that this extended BFM was capable of performing simulations of polymer systems with increased efficiency we focused on a particular system of interest. For many technical applications it is important to have a knowledge of how copolymers

E. James, CC. Matthai/Computational

Materials Science 10 (1998) 175-I 79

177

Bond Length Spsctra

0.3

I

I

,

I

I

I

NewModel

,-\ ,/*

,_I’

-

\\

__,’

\\

%\.

Old Model ----

0.25

oz $ z F LL .o ‘-I E z

0.2 -

0.15

-

,’ 1’ : ,’ ,’ : ,’ 1’ ,’

0.1 : /’

l 0.05 -

8’

/ OL1

1.1

1.2

1.3 1.4 Normalised Bond Length

1.5

1.6

1.7

Fig. 1. Norma&d bond length distributions of a single polymer chain using the original and modified BFM. The peaks correspond to the recorded bond lengths.

I

0.12

I

Bond Angle Spectra

I

I

I

I

1 New Model ~ Old Model ----.

0.1

E

0.06

-

0.06

-

0.04

-

5 g 1 E b 2

0.02 -

n 0

0.1

0.2

0.3

ino&its Angle

0.5

0.6

0.7

0.6

of Pi

Fig. 2. Bond angle distributions of a single polymer chain using the original and modified BFh4. The peaks correspond to the recorded angles.

E. James, C.C. Matthai/Computational

178

Materials Science 10 (1998) 175-l 79

Z Monomer Distributions about the interface for Chains of Length 128

I

0.25

I

I

I

!

I

I

A Monomers B Monomers ~~~~

1

II

0.2

ox 5

ii

0.15

s a, II TJ w c g

0.1

Z

0.05

0 c -5 0 5 Z Distance from the Interface Fig.

3. The

normalised

at an interface.

Note

distribution

of A (full)

that the different

peak

and B (dashed) heights

are

type monomers

for a copolymer

due to the different number of

A

chain of length

and

B type

N

=

monomers

128% in the

polymer.

behave

at the interface

have therefore

between

used the extended

two

solvents.

We

BFM to perform

for,- i :L,

E(A) = -xknT,

forz > ;L,

E(A) = 0,

E(B) = 0,

E(B) = -xkBT.

MC simulations on random copolymers consisting of N randomly chosen A and B monomer units placed

Moves were then considered

according

above and the usual statistical

to the rules

at the interface of two solvents, one of which favours

outlined

the A-type monomers and the other preferring B-type monomers. These simulations are similar to those per-

factor

ject otherwise

formed by Peng et al. [3] who employed the original BFM. The simulations were performed on single random

chains were placed in random configurations, with no attempt to constrain the numbers of the A and B types to be the same, and the system allowed

copolymer chains of lengths, N = 16,32,64 and 128 units which were placed in a (L x L x L) box with L =

to equilibrate (200000 moves) after which the z co-ordinates of the monomers of each type were

6N. This ensured that the box was large enough to prevent the chain from interacting with itself through the periodic boundaries which were imposed in the (Xy) plane of the interface. Fixed boundary conditions were imposed at z = 0 and z = L. The interactions between the polymer and the solvent were modelled by prescribing the energies, E(A) and E(B), of the A and B monomers as follows:

recorded. For each polymer, we performed 24 separate simulations. Typical z-distributions are shown in Fig. 3. The results can be seen to be very similar to that obtained by Peng et al. Interestingly, the secondary peak in each distribution function, which is found in the environment of the unfavourable solvent, and is associated with frustration, persists even in our finer grain model suggesting that

criterion

was

employed

energetically

Boltzmann

to accept

forbidden

or re-

moves.

The

E. James, CC. Matthai/Computational

it is not an artefact of the lattice structure of the simulations.

Materials Science IO (1998) 175-l 79

179

Although the BFM method may not be very efficient in generating static configurations, it is able to give dynamical information which is almost impossible with other Monte Carlo methods.

5. Conclusion By extending the BFM, we have demonstrated that there may be some optimum lattice mesh in coarsegrained model simulations which, with little or no increase in computation time, can give a more detailed structure of the polymer systems being simulated. We tested the 8 and 27 site models on a simple single chain system and found that computational efficiency was increased with the new 27 site model. We have used this model to simulate the random copolymer configuration at an interface of two solvents and found that the results are in good agreement with earlier work.

Acknowledgements

EJ acknowledges financial support from the University of Wales, Cardiff.

References [l] I. Carmesin, K. Kremer, Macromolecules 12 (1988) 2819. [2] H.P. Deutsch, K. Binder, J. Chem. Phys. 94 (1991) 2294. [3] G. Peng, J.U. Sommer, A. Blumen, Phys. Rev. E 53 (1996) 5509.